Republicans have resisted providing money to the states — what the Senate majority leader, Mitch McConnell of Kentucky, called “blue state bailouts” — even as local governments have been overwhelmed by an explosion of unemployment claims, with more than 26 million people losing their jobs in just five weeks.

At least three states — California, New York and Ohio — are expected to deplete their trust funds within two weeks, with Massachusetts, Texas and Mr. McConnell’s state of Kentucky close behind. Once those funds run out, the states can borrow money from the federal government, but must repay it within two years.

Delays in delivering benefits, though, are as troubling as the sheer magnitude of the figures. Such problems not only create immediate hardships, like not being able to pay rent or buy food, but also affect the shape of the recovery when the pandemic eases.

In New York City, about 21 percent tested positive for coronavirus antibodies during the state survey. The rate was about 17 percent on Long Island, nearly 12 percent in Westchester and Rockland Counties and less than 4 percent in the rest of the state.

In January, a mystery illness swept through a call center in a skyscraper in Chicago. Close to 30 people in one department alone had symptoms — dry, deep coughs and fevers they could not shake. When they gradually returned to work after taking sick days, they sat in their cubicles looking wan and tired.

“I’ve started to think it was the coronavirus,” said Julie Parks, a 63-year-old employee who was among the sick. “I may have had it, but I can’t be sure.”

The retroactive search is happening on many levels. People who had suffered dreadful bouts with flulike illnesses are now wondering if it was the coronavirus. Doctors are thinking back to unexplained cases. Medical examiners are poring over their records looking for possible misdiagnosed deaths. And local politicians are demanding investigations.

“I think it was here long before we knew it,” said Brian Gustafson, a coroner in Rock Island County, Ill. “That’s the only logical thing I can think of.”

After the scientist, William N. Bryan, the head of science at the Department of Homeland Security, told the briefing that the government had tested how sunlight and disinfectants — including bleach and alcohol — could kill the coronavirus on surfaces in as little as 30 seconds, an excited Mr. Trump returned to the lectern.

“Supposing we hit the body with a tremendous — whether it’s ultraviolet or just very powerful light,” Mr. Trump said. “And I think you said that hasn’t been checked, but we’re going to test it?” he added, turning to Mr. Bryan, who had returned to his seat. “And then I said, supposing you brought the light inside the body, either through the skin or some other way.”

Apparently reassured that the tests he was proposing would take place, Mr. Trump then theorized about the possible medical benefits of disinfectants in the fight against the virus.

“And then I see the disinfectant where it knocks it out in a minute — one minute — and is there a way we can do something like that by injection inside, or almost a cleaning?” he asked. “Because you see it gets in the lungs and it does a tremendous number on the lungs, so it would be interesting to check that.”

Experts have long warned that ultraviolet lamps can harm humans if used improperly — when the exposure is outside the body, much less inside. But bottles of bleach and other disinfectants carry sharp warnings of ingestion dangers. The disinfectants can kill not only microbes but also humans.

Rather than looking for handouts, Mr. McConnell said, the states should consider filing for bankruptcy. His aides threw fuel on the fire in a news release that said the Senate leader was opposed to “blue state bailouts,” suggesting it was Democratic-leaning states that were seeking the money to take care of problems caused by fiscal mismanagement.

“That’s how you’re going to bring this national economy back?” asked an incredulous Mr. Cuomo, who called Mr. McConnell irresponsible and reckless. “You want to see that market fall through the cellar? Let New York State declare bankruptcy.”

Republicans on Capitol Hill say they believe that Mr. McConnell, who opposed new state aid in talks that produced the most recent measure, was trying to reassure restive conservatives that he would not give in easily on more funding in coming talks after Congress had already allocated about $2.7 trillion in deficit spending in response to the emergency. But Mr. McConnell faces significant obstacles if he intends to block the aid, given the extent of bipartisan support for more state relief.

The result is that governors across the country, even those allied with Mr. Trump, are all but forced to pay close attention to the administration’s guidance on the timing of opening up their economies. And that guidance, critics say, is all over the place.

“You know you’re going to be left hanging out to dry if you make a call that’s at odds with Trump’s psyche or mood or thinking on a given day,” said Mark Sanford, former Republican governor of South Carolina and a persistent critic of Mr. Trump. “And I think that in political terms, given the size of his base, that adds a level of complexity, particularly for red-state governors.”

Mr. Kemp’s order also allows nail salons, gyms, bowling alleys and tattoo parlors to reopen on Friday. Dine-in service at restaurants will be allowed to resume on Monday.

But even as some hairstylists were readying work spaces for their first customers in weeks on Friday morning in Georgia, others said they would stay home, afraid of spreading the coronavirus to clients.

It remains to be seen how broad the buy-in will be. In Atlanta, the hair stylist Lindsey Maxfield, 33, said she was glad her workplace, Cameo Salon, would remain closed. “Having people come to the salon is ridiculous,” she said.

Even as the European Center for Disease Prevention and Control announced that the first wave of coronavirus transmission had “passed its peak” in 20 members of the European Union, Britain was still struggling to get ahead of the virus.

The nation was behind many other countries in Europe in putting in place restrictive social distancing measures, with the British government frequently saying it was “guided by the science.” With the country approaching 20,000 deaths, the Times correspondents Mark Landler and Stephen Castle took a look at the secretive scientific group advising the government.

As the British government comes under mounting criticism for its response to the coronavirus — one that has left Britain vying with Italy and Spain as the worst hit countries in Europe — Prime Minister Boris Johnson and his aides have defended themselves by saying they are “guided by the science.”

The trouble is, nobody knows what the science is.

The government’s influential Scientific Advisory Group for Emergencies — known by its soothing acronym, SAGE — operates as a virtual black box. Its list of members is secret, its meetings are closed, its recommendations are private and the minutes of its deliberations are published much later, if at all.

Yet officials invoke SAGE’s name endlessly without ever explaining how it comes up with its advice — or even who these scientists are.

That lack of transparency has become a point of contention, as officials struggle to explain why they waited until late March to shift from a laissez-faire approach to the virus to the stricter measures adopted by other European countries. Critics say the delay may have worsened a death toll now surging past 20,000, and they fault the government for leaving people in the dark about why it first chose this riskier path.

What else is happening around the world.

Keep up with developments in the coronavirus pandemic with our team of international correspondents.